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Let $q$ be a prime power and $d$ an integer. I want to understand the classification of the transitive linear subgroups of $GL_d(\mathbb F_q)$. According to the Wikipedia page https://en.wikipedia.org/wiki/List_of_transitive_finite_linear_groups these fall into finitely many classes, four of which are infinite, plus finitely many sporadic cases. I have several questions about this list though, because it is unclear to me how should one read it.

1) Does one have to read it in the following way: "if $G\leq GL_d(\mathbb F_p)$ acts transitively on $\mathbb F_p^d$ then it belongs to one of these classes"? because in this case, it seems to me that the second row with $a=1$ reads "the trivial subgroups is normal in $G$", which is of course a trivial statement. Clearly I am misunderstanding something, could someone please clarify this to me?

2) What sort of group is $\Gamma L(1,p^d)$?

3) Does that list change in any serious way if I consider general finite fields $\mathbb F_{p^n}$ instead of $\mathbb F_p$?

4) Is there any clean reference that reports a table of transitive subgroups of $GL_d(\mathbb F_p)$?

Thanks a lot in advance!

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    $\begingroup$ For 1), I expect the second row should say "for $a>1$". For 2), ${\rm \Gamma L}(a,q^d)$ is the extension of ${\rm GL}(a,q^d)$ with a field automorphism of order $d$. For 3), note that a transitive subgroup of ${\rm GL}(d,p^n)$ could be regarded as a transitive subgroup of ${\rm GL}(dn,p)$, which might help. $\endgroup$
    – Derek Holt
    Commented Jul 5, 2019 at 11:34
  • $\begingroup$ To answer (4) you could look at the appendix of Liebeck's "The affine permutation groups of rank three" where he reproves Hering's theorem. I have an e-copy of this paper -- email me if you want me to send it to you. $\endgroup$
    – Nick Gill
    Commented Jul 8, 2019 at 12:39
  • $\begingroup$ Thanks a lot for the comments! @NickGill: I have found a copy of the paper myself, thanks for the reference! $\endgroup$
    – Ferra
    Commented Jul 8, 2019 at 17:51

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